
diff -r 20de3d710f77 -r ae1ee41f20dd pnas/pnas.tex
--- a/pnas/pnas.tex	Thu Nov 11 15:48:47 2010 -0800
+++ b/pnas/pnas.tex	Thu Nov 11 17:50:28 2010 -0800
@@ -199,7 +199,7 @@
 We will treat each of these in turn.
 
 To motivate our morphism axiom, consider the venerable notion of the Moore loop space
-\nn{need citation -- \S 2.2 of Adams' ``Infinite Loop Spaces''?}.
+\cite[\S 2.2]{MR505692}.
 In the standard definition of a loop space, loops are always parameterized by the unit interval $I = [0,1]$,
 so composition of loops requires a reparameterization $I\cup I \cong I$, and this leads to a proliferation
 of higher associativity relations.
@@ -448,13 +448,15 @@
 with coefficients in the $n$-category $\cC$ as the homotopy colimit along $\cell(W)$
 of the functor $\psi_{\cC; W}$ described above. We write this as $\clh{\cC}(W)$.
 
-An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{x} \psi_{\cC; W}(x_0)[m],$$ where $x = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on \todo{finish this}
+An explicit realization of the homotopy colimit is provided by the simplices of the functor $\psi_{\cC; W}$. That is, $$\clh{\cC}(W) = \DirectSum_{\bar{x}} \psi_{\cC; W}(x_0)[m],$$ where $\bar{x} = x_0 \leq \cdots \leq x_m$ is a simplex in $\cell(W)$. The differential acts on $(\bar{x},a)$ (here $a \in \psi_{\cC; W}(x_0)$) as
+$$\bdy (\bar{x},a) = (\bar{x}, \bdy a) + (-1)^{\deg a} \left( (d_0 \bar{x}, g(a)) + \sum_{i=1}^m (-1)^i (d_i \bar{x}, a) \right)$$
+where $g$ is the gluing map from $x_0$ to $x_1$, and $d_i \bar{x}$ denotes the $i$-th face of the simplex $\bar{x}$.
 
-Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
+Alternatively, we can take advantage of the product structure on $\cell(W)$ to realize the homotopy colimit as the cone-product polyhedra of the functor $\psi_{\cC;W}$. (A cone-product polyhedra is obtained from a point by successively taking the cone or taking the product with another cone-product polyhedron; just as simplices correspond to linear graphs, cone-product polyheda correspond to directed trees.) A Eilenberg-Zilber subdivision argument shows this is the same as the usual realization.
 
 When $\cC$ is a topological $n$-category,
 the flexibility available in the construction of a homotopy colimit allows
-us to give a much more explicit description of the blob complex. We'll write $\bc_*(W; \cC)$ for this more explicit version.
+us to give a much more explicit description of the blob complex which we'll write as $\bc_*(W; \cC)$.
 
 We say a collection of balls $\{B_i\}$ in a manifold $W$ is \emph{permissible}
 if there exists a permissible decomposition $M_0\to\cdots\to M_m = W$ such that
@@ -549,7 +551,7 @@
 H_0(\bc_*(X;\cC)) \iso A_{\cC}(X)
 \end{equation*}
 \end{thm}
-This follows from the fact that the $0$-th homology of a homotopy colimit of \todo{something plain} is the usual colimit, or directly from the explicit description of the blob complex.
+This follows from the fact that the $0$-th homology of a homotopy colimit is the usual colimit, or directly from the explicit description of the blob complex.
 
 \begin{thm}[Hochschild homology when $X=S^1$]
 \label{thm:hochschild}
@@ -630,7 +632,7 @@
 \label{thm:product}
 Let $W$ be a $k$-manifold and $Y$ be an $n-k$ manifold.
 Let $\cC$ be an $n$-category.
-Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ via blob homology.
+Let $\bc_*(Y;\cC)$ be the $A_\infty$ $k$-category associated to $Y$ as above.
 Then
 \[
 	\bc_*(Y\times W; \cC) \simeq \clh{\bc_*(Y;\cC)}(W).
@@ -639,7 +641,7 @@
 The statement can be generalized to arbitrary fibre bundles, and indeed to arbitrary maps
 (see \cite[\S7.1]{1009.5025}).
 
-Fix a topological $n$-category $\cC$, which we'll omit from the notation.
+Fix a topological $n$-category $\cC$, which we'll now omit from notation.
 Recall that for any $(n-1)$-manifold $Y$, the blob complex $\bc_*(Y)$ is naturally an $A_\infty$ category.
 
 \begin{thm}[Gluing formula]
@@ -688,7 +690,7 @@
 By the `blob cochains' of a manifold $X$, we mean the $A_\infty$ maps of $\bc_*(X)$ as a $\bc_*(\bdy X)$ $A_\infty$-module.
 
 \begin{proof}
-We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and functoriality.
+We have already defined the action of mapping cylinders, in Theorem \ref{thm:evaluation}, and the action of surgeries is just composition of maps of $A_\infty$-modules. We only need to check that the relations of the $n$-SC operad are satisfied. This follows immediately from the locality of the action of $\CH{-}$ (i.e., that it is compatible with gluing) and associativity.
 \end{proof} 
 
 The little disks operad $LD$ is homotopy equivalent to the $n=1$ case of the $n$-SC operad. The blob complex $\bc_*(I, \cC)$ is a bimodule over itself, and the $A_\infty$-bimodule intertwiners are homotopy equivalent to the Hochschild cohains $Hoch^*(C, C)$. The usual Deligne conjecture (proved variously in \cite{hep-th/9403055, MR1805894, MR2064592, MR1805923}) gives a map




